Equilibrium stresses in frameworks via symmetric averaging

Abstract For a bar-joint framework $$(G,\textbf{p})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , a subgroup $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> of the automorphism group of G , and a subgroup of the orthogonal group isomorphic to $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> , we introduce a symmetric averaging map which produces a bar-joint framework on G with that symmetry. If the original configuration is “almost symmetric”, then the averaged one will be near the original configuration. With a view on structural engineering applications, we then introduce a hierarchy of definitions of “localised” and “non-localised” or “extensive” self-stresses of frameworks and investigate their behaviour under the symmetric averaging procedure. Finally, we present algorithms for finding non-degenerate symmetric frameworks with many states of self-stress, as well as non-symmetric and symmetric frameworks with extensive self-stresses. The latter uses the symmetric averaging map in combination with symmetric Maxwell-type character counts and a procedure based on the pure condition from algebraic geometry. These algorithms provide new theoretical and computational tools for the design of engineering structures such as gridshell roofs.